There are several ways that this derivative information can be obtained. For sufficiently simple figures of merit, the objective can be analytically differentiated and implemented by hand. Alternatively, the derivatives can be obtained programmatically using automatic differentiation tools. When a given objective function depends on the solution of a set of equations, such as the MHD equilibrium equations (), the derivatives can be obtained more efficiently using an adjoint method. With this technique, the solution of only one additional equation, known as the adjoint equation, is required. Once the adjoint solution is obtained, the derivative of a given objective can be obtained with respect to any optimization parameter, eliminating the need to solve a perturbed set of equations. In this way, the cost associated with obtaining a high-dimensional gradient is significantly reduced. In this work, we employ adjoint-based gradients of functions which depend on the MHD equilibrium equations. This adjoint method results from a generalized self-adjointness property of the MHD force operator [1]}.
This technique has been demonstrated for computing the shape derivatives of several figures of merit relevant for stellarator design, including the magnetic well, rotational transform, and magnetic ripple [2]}. We also employ analytic gradient for objectives that do not depend on the MHD equilibrium equations, such as the volume and properties of the surface curvatures.
